Exclusive production of ηπ − and η π − has been studied with a 191 GeV/c π − beam impinging on a hydrogen target at COMPASS (CERN). Partial-wave analyses reveal different odd/even angular momentum (L) characteristics in the inspected invariant mass range up to 3 GeV/c 2 . A striking similarity between the two systems is observed for the L = 2, 4, 6 intensities (scaled by kinematical factors) and the relative phases. The known resonances a 2 (1320) and a 4 (2040) are in line with this similarity. In contrast, a strong enhancement of η π − over ηπ − is found for the L = 1, 3, 5 waves, which carry non-qq quantum numbers. The L = 1 intensity peaks at 1.7 GeV/c 2 in η π − and at 1.4 GeV/c 2 in ηπ − , the corresponding phase motions with respect to L = 2 are different.